We study the problem of auction design for advertising platforms that face strategic advertisers who are bidding across platforms. Each advertiser's goal is to maximize their total value or conversions while satisfying some constraint(s) across all the platforms they participates in. In this paper, we focus on advertisers with return-over-investment (henceforth, ROI) constraints, i.e. each advertiser is trying to maximize value while making sure that their ROI across all platforms is no less than some target value. An advertiser interacts with the platforms through autobidders -- for each platform, the advertiser strategically chooses a target ROI to report to the platform's autobidder, which in turn uses a uniform bid multiplier to bid on the advertiser's behalf on the queries owned by the given platform.

Multi-item revenue-optimal mechanisms are known to be extremely complex, often offering buyers randomized lotteries of goods. In the standard buy-one model, it is known that optimal mechanisms can yield revenue infinitely higher than that of any "simple" mechanism---the ones with size polynomial in the number of items---even with just two items and a single buyer [Briest et al. 2015; Hart and Nisan 2017].

We consider a revenue-maximizing seller with $k$ heterogeneous items for sale to a single additive buyer, whose values are drawn from a known, possibly correlated prior $\mathcal{D}$. It is known that there exist priors $\mathcal{D}$ such that simple mechanisms --- those with bounded menu complexity --- extract an arbitrarily small fraction of the optimal revenue~(Briest et al. 2015, Hart and Nisan 2019). This paper considers the opposite direction: given a correlated distribution $\mathcal{D}$ witnessing an infinite separation between simple and optimal mechanisms, what can be said about $\mathcal{D}$?\citet{hart2019selling} provides a framework for constructing such $\mathcal{D}$: it takes as input a sequence of $k$-dimensional vectors satisfying some geometric property, and produces a $\mathcal{D}$ witnessing an infinite gap. Our first main result establishes that this framework is without loss: every $\mathcal{D}$ witnessing an infinite separation could have resulted from this framework. An earlier version of their work provided a more streamlined framework (Hart and Nisan 2013). Our second main result establishes that this restrictive framework is not tight. That is, we provide an instance $\mathcal{D}$ witnessing an infinite gap, but which provably could not have resulted from the restrictive framework. As a corollary, we discover a new kind of mechanism which can witness these infinite separations on instances where the previous ``aligned'' mechanisms do not.

We consider the manipulability of tournament rules, in which $n$ teams play a round robin tournament and a winner is (possibly randomly) selected based on the outcome of all $\binom{n}{2}$ matches. Prior work defines a tournament rule to be $k$-SNM-$α$ if no set of $\leq k$ teams can fix the $\leq \binom{k}{2}$ matches among them to increase their probability of winning by $>α$ and asks: for each $k$, what is the minimum $α(k)$ such that a Condorcet-consistent (i.e. always selects a Condorcet winner when one exists) $k$-SNM-$α(k)$ tournament rule exists? A simple example witnesses that $α(k) \geq \frac{k-1}{2k-1}$ for all $k$, and [Schneider et al., 2017] conjectures that this is tight (and prove it is tight for $k=2$). Our first result refutes this conjecture: there exists a sufficiently large $k$ such that no Condorcet-consistent tournament rule is $k$-SNM-$1/2$. Our second result leverages similar machinery to design a new tournament rule which is $k$-SNM-$2/3$ for all $k$ (and this is the first tournament rule which is $k$-SNM-$(<1)$ for all $k$). Our final result extends prior work, which proves that single-elimination bracket with random seeding is $2$-SNM-$1/3$([Schneider et al., 2017]), in a different direction by seeking a stronger notion of fairness than Condorcet-consistence. We design a new tournament rule, which we call Randomized-King-of-the-Hill, which is $2$-SNM-$1/3$ and \emph{cover-consistent} (the winner is an uncovered team with probability $1$).

We consider optimal (revenue maximizing) mechanism design in the interdimensional setting, where one dimension is the 'value' of the buyer, and the other is a 'type' that captures some auxiliary information. A prototypical example of this is the FedEx Problem, for which Fiat et al. [2016] characterize the optimal mechanism for a single agent. Another example of this is when the type encodes the buyer's budget [DW17]. The question we address is how far can such characterizations goIn particular, we consider the setting of single-minded agents. A seller has heterogenous items. A buyer has a valuation vfor a specific subset of items S, and obtains value vif and only if he gets all the items in S(and potentially some others too).

We consider a revenue-maximizing seller with n items facing a single buyer. We introduce the notion of symmetric menu complexity of a mechanism, which counts the number of distinct options the buyer may purchase, up to permutations of the items. Our main result is that a mechanism of quasi-polynomial symmetric menu complexity suffices to guarantee a (1 - epsilon )-approximation when the buyer is unit-demand over independent items, even when the value distribution is unbounded, and that this mechanism can be found in quasi-polynomial time. Our key technical result is a polynomial-time, (symmetric) menu-complexity-preserving black-box reduction from achieving a (1 - epsilon )-approximation for unbounded valuations that are subadditive over independent items to achieving a (1 - O(epsilon ))-approximation when the values are bounded (and still subadditive over independent items). We further apply this reduction to deduce approximation schemes for a suite of valuation classes beyond our main result. Finally, we show that selling separately (which has exponential menu complexity) can be approximated up to a (1 - epsilon ) factor with a menu of efficient-linear (f (epsilon) · n) symmetric menu complexity.

We study the menu complexity of optimal and approximately-optimal auctions in the context of the “FedEx” problem, a so-called “one-and-a-half-dimensional” setting where a single bidder has both a value and a deadline for receiving an item [FGKK16]. The menu complexity of an auction is equal to the number of distinct (allocation, price) pairs that a bidder might receive [HN13]. We show the following when the bidder has n possible deadlines: Exponential menu complexity is necessary to be exactly optimal: There exist instances where the optimal mechanism has menu complexity ≥ 2n – 1. This matches exactly the upper bound provided by Fiat et al.'s algorithm, and resolves one of their open questions [FGKK16]. Fully polynomial menu complexity is necessary and sufficient for approximation: For all instances, there exists a mechanism guaranteeing a multiplicative (1 – ∊)-approximation to the optimal revenue with menu complexity , where υmax denotes the largest value in the support of integral distributions. There exist instances where any mechanism guaranteeing a multiplicative (1 – O(1/n2))-approximation to the optimal revenue requires menu complexity Ω(n2). Our main technique is the polygon approximation of concave functions [Rot92], and our results here should be of independent interest. We further show how our techniques can be used to resolve an open question of [DW17] on the menu complexity of optimal auctions for a budget-constrained buyer.

While it is known that using network coding can significantly improve the throughput of directed networks, it is a notorious open problem whether coding yields any advantage over the multicommodity flow (MCF) rate in undirected networks. It was conjectured that the answer is no. In this paper we show that even a small advantage over MCF can be amplified to yield a near-maximum possible gap. We prove that any undirected network with k source-sink pairs that exhibits a (1+epsilon) gap between its MCF rate and its network coding rate can be used to construct a family of graphs G' whose gap is log(|G'|)^c for some constant c < 1. The resulting gap is close to the best currently known upper bound, log(|G'|), which follows from the connection between MCF and sparsest cuts. Our construction relies on a gap-amplifying graph tensor product that, given two graphs G1,G2 with small gaps, creates another graph G with a gap that is equal to the product of the previous two, at the cost of increasing the size of the graph. We iterate this process to obtain a gap of log(|G'|)^c from any initial gap.

We consider the manipulability of tournament rules for round-robin tournaments of n competitors. Specifically, n competitors are competing for a prize, and a tournament rule r maps the result of all n(n-1)/2 pairwise matches (called a tournament, T) to a distribution over winners. Rule r is Condorcet-consistent if whenever i wins all n-1 of her matches, r selects i with probability 1. We consider strategic manipulation of tournaments where player j might throw their match to player i in order to increase the likelihood that one of them wins the tournament. Regardless of the reason why j chooses to do this, the potential for manipulation exists as long as Pr[r(T) = i] increases by more than Pr[r(T) = j] decreases. Unfortunately, it is known that every Condorcet-consistent rule is manipulable. In this work, we address the question of how manipulable Condorcet-consistent rules must necessarily be - by trying to minimize the difference between the increase in Pr[r(T) = i] and decrease in Pr[r(T) = j] for any potential manipulating pair. We show that every Condorcet-consistent rule is in fact 1/3-manipulable, and that selecting a winner according to a random single elimination bracket is not alpha-manipulable for any alpha > 1/3. We also show that many previously studied tournament formats are all 1/2-manipulable, and the popular class of Copeland rules (any rule that selects a player with the most wins) are all in fact 1-manipulable, the worst possible. Finally, we consider extensions to match-fixing among sets of more than two players.